On the spread of epidemics in a closed heterogeneous population

TL;DR

This paper explores how heterogeneity in populations affects epidemic spread, deriving a nonlinear transmission function from heterogeneous models, and highlights potential pitfalls of traditional approximation methods.

Contribution

It provides a mechanistic derivation of nonlinear transmission functions from heterogeneous population models, connecting them to phenomenological models.

Findings

Heterogeneous models can be reduced to homogeneous models with nonlinear transmission functions.

The power transmission function is derived from gamma-distributed parameters.

Standard moment-closure methods may lead to errors in long-term epidemic predictions.

Abstract

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed for different sources of heterogeneity. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from a heterogeneous model with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which mimics reality very well, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the model.